Let \(L\)  be a Legendrian knot in \((M,\xi ^M_{\textrm {stab}})\)  such that \(L(-1)=(N,\xi ^N)\) . Let \(L^*\)  be the dual surgery knot of \(L\)  in \((N,\xi ^N)\) . By the cancellation lemma, \(L^*(+1)\)  is again contactomorphic to \((M,\xi ^M_{\textrm {stab}})\) . Now we choose a loose Legendrian realization \(L^{\prime }\)  of \(L^*\)  such that if we stabilize \(L^{\prime }\)  once positive and once negative we get a Legendrian knot which is formally isotopic to \(L^*\) . (This is possible since we can destabilize any loose Legendrian knot.)
We claim that \(L^{\prime }(-1)\)  yields \((M,\xi ^M)\) . Since \(L^{\prime }\)  is topologically isotopic to \(L^*\)  and its contact framing and the contact framing of \(L^*\)  differ by 2, the contact \((-1)\) -surgery along \(L^{\prime }\)  yields topologically the same manifold as the contact \((+1)\) -surgery along \(L^*\)  which is in fact \(M\) . Note that \(L^{\prime }(-1)\)  is overtwisted since \(L^{\prime }\)  is a loose knot. Then a straightforward computation in a local model as in [1]} shows that the homotopical invariants of the contact structures agree. Thus by Eliashberg's classification of contact structures [2]} it follows that the contact structures are contactomorphic.
